Variogram

Lack of accurate climate data is a problem that facing many Arab countries including Libya. This problem affects integrated planning on the environment and the agriculture. There is an increasing demand for gridded datasets of climate variables from fields such as hydrology, ecology, agriculture, climate change research and climate model verification. Consequently this paper attempted to make at spatial interpolation, of main climatic parameters using spatial geostatistics tools. This paper gave a brief overview of how to produce accurate climatic maps by producing a consistent series of climatic statistics which enables comparisons to be made across space and time using contemporary techniques that are inherent in the Science of Geographic Information Systems (GIS) and Genstat (Ordinary Kriging). The maps were created using Ordinary Kriging with accuracy of estimates for each 1000 m x 1000 m pixel. The kriged map for rainfall shows a rapid increase in mean annual rainfall from the ...

Traditional geostatistical estimation relies on a stationary variogram within
chosen rock types or spatial domains; however, there are often locationdependent
variations across the chosen domain. Subdividing the domain is not
always practical; the number of data becomes too few for reliable inference.
Moreover, changes in the spatial behaviour of the variable can occur smoothly
across the domain. Location dependent variograms based on spatial weights are
proposed to depict the local spatial variability. The weights applied to the
sample pairs are obtained by a Gaussian Kernel function. The location
dependent experimental variograms are semi-automatically modelled to yield
local variogram parameters. These parameters define the local anisotropy
orientation, ranges, nugget effect, and variogram shape. They are smoothly
interpolated at the resolution of estimation and subsequently used in a quasistationary
kriging approach. The same variogram is considered for all distances
within the search radius centered at the location being estimated. This assures
the positive definiteness of the kriging equations. The improvement in local
estimation is significant. Local features are better reproduced in estimation,
while the smoothing effect is mitigated. The kriging variances provide a more
complete measure of local accuracy.

Rock mass strength is typically estimated from three fundamental components: intact rock strength, fracture strength, and intensity of fracturing. The uncertainties regarding the estimation of rock mass shear strength can also be separated into three distinct components: natural variability, statistical uncertainty, and transformation uncertainty. The natural spatial variability of geologic materials typically has the greatest impact on the uncertainty of rock mass design parameters. If the spatial continuity (autocorrelation) of input variables is not considered, a probabilistic slope analysis can result in either over-or underestimation of the probability of failure. It is well established that the stability of a slope is controlled by the total shear resistance along the failure surface, rather than the shear resistance at any one individual location. For probabilistic slope analysis, it is therefore appropriate to determine the variance of the rock mass strength over the entire potential zone of failure. The variance of the shear strength across the failure surface will always be less than the variance of individual small-scale measurements of shear strength at any one location on that surface. The scale of fluctuation of each of the three components of rock mass strength (intact strength, fracture strength, intensity of fracturing) must be considered in order to estimate the variance of rock mass strength over the failure surface area. Intact rock strength and fracture strength typically exhibit a small spatial correlation distance relative to typical open pit slope analysis. Therefore, it is proposed that the natural variability for these parameters can be disregarded and in most cases only the statistical uncertainty of intact and fracture shear strength derived from the total population of representative laboratory tests need to be considered. Conversely, intensity of fracturing, as measured by either RQD or the GSI structure rating, tends to exhibit large spatial correlation distances relative to typical sampling intervals. A variogram model can be used in conjunction with the variance reduction function to characterize the spatial variability of the fracture intensity and to quickly estimate the reduced variance due to spatial averaging for use in probabilistic analysis. Examples are provided to demonstrate the value of these concepts.

We analyzed spatial and temporal variograms of precipitation, runoff, and groundwater levels in Austria to examine whether characteristic scales exist and, if so, how big they are. In time, precipitation and runoff are stationary with characteristic scales on the order of a day and a month, respectively, while groundwater levels are nonstationary. In space, precipitation is almost fractal, so no characteristic scales exist. Runoff is nonstationary but not a fractal as it exhibits a break in the variograms. An analysis of the variograms of catchment precipitation indicates that this break is due to aggregation effects imposed by the catchment area. A spatial variogram of hypothetical point runoff back calculated from runoff variograms of three catchment size classes using aggregation statistics (regularization) is almost stationary and exhibits shorter characteristic space scales than catchment runoff. Groundwater levels are nonstationary in space, exhibiting shorter-scale variability than precipitation and runoff, but are also not fractal as there is a break in the variogram. We suggest that the decrease of spatial characteristic scales from catchment precipitation to runoff and to groundwater is the result of a superposition of small-scale variability of catchment and aquifer properties on the rainfall forcing. For comparison, TDR soil moisture data from a comprehensive Australian data set were examined. These data suggest that in time, soil moisture is close to stationary with characteristic scales of the order of 2 weeks while in space soil moisture is nonstationary and close to fractal over the extent sampled. Space-time traces of characteristic scales fit well into a conceptual diagram of Blöschl and Sivapalan [1995] . The scaling exponents z in T ∼ L z (where T is time and L is space) are of the order of 0.5 for precipitation, 0.8 for runoff from small catchments, 1.2 for runoff from large catchments, 0.8 for groundwater levels, and 0.5 for soil moisture.

Variogram estimation plays an important role in many areas of spatial statistics. Potential areas of application include biology, ecology, economics and meteorology. However, it is common that, for example under highly correlated patterns, traditional estimators can not reflect all the spatial features or dependencies. In this paper, we present an alternative distribution-free estimator based on nearest-neighbour estimation with a non-constant smoothing field that is better able to adapt to spatially varying features of the data pattern. We present a simulation study to compare our new estimator to a nearest-neighbour estimator built with a constant smoothing parameter and to the classical variogram estimator. We apply our method to analyze two ecological data sets.

Recent research has expanded our understanding of microbial community assembly. However, the field of community ecology is inaccessible to many microbial ecologists because of inconsistent and often confusing terminology as well as unnecessarily polarizing debates. Thus, we review recent literature on microbial community assembly, using the framework of Vellend (Q. Rev. Biol. 85:183–206, 2010) in an effort to synthesize and unify these contributions. We begin by discussing patterns in microbial biogeography and then describe four basic processes (diversification, dispersal, selection, and drift) that contribute to community assembly. We also discuss different combinations of these processes and where and when they may be most important for shaping microbial communities. The spatial and temporal scales of microbial community assembly are also discussed in relation to assembly processes. Throughout this review paper, we highlight differences between microbes and macroorganisms and generate hypotheses describing how these differences may be important for community assembly. We end by discussing the implications of microbial assembly processes for ecosystem function and biodiversity.

SUMMARY Recent research has expanded our understanding of microbial community assembly. However, the field of community ecology is inaccessible to many microbial ecologists because of inconsistent and often confusing terminology as well as unnecessarily polarizing debates. Thus, we review recent literature on microbial community assembly, using the framework of Vellend (Q. Rev. Biol. 85 :183–206, 2010) in an effort to synthesize and unify these contributions. We begin by discussing patterns in microbial biogeography and then describe four basic processes (diversification, dispersal, selection, and drift) that contribute to community assembly. We also discuss different combinations of these processes and where and when they may be most important for shaping microbial communities. The spatial and temporal scales of microbial community assembly are also discussed in relation to assembly processes. Throughout this review paper, we highlight differences between microbes and macroorganism...

The modern mining industry employs plenty of exploration data digitization and utilizes computational resources for future forecasting and production scheduling. In that regard, geostatistics and mine planning as disciplines are critical parts of the mining business. However, traditional mine planning does not allow the risk management associated with geological uncertainty due to using a single mineral resource model as an input. Integration of stochastic geostatistical realizations into mine planning can help to minimize the risk of not meeting the production targets. Nevertheless, there are still many limitations in commercially used stochastic geostatistical algorithms, particularly multivariate methods. For example, it is common to deal with challenging datasets containing bivariate complexities among variables, such as inequality constraints. The poor reproduction of this feature can lead to an overestimation of the secondary variable, affecting the processing cost. Therefore, this work's motivation is to propose a way to model datasets containing inequality constraints between variables and generate production schedules considering geological uncertainty. 
This study proposes an algorithm based on a hierarchical cosimulation framework integrated with inverse transform sampling, which is designed to model variables within thresholds derived from a linear inequation. The proposed algorithm is applied to a real case study from an Iron deposit with a sharp inequality constraint between Iron and Silica. Results are assessed based on validation of marginal distribution, spatial correlation and bivariate relationship. Overall, the proposed algorithm demonstrates simulations of similar quality as the conventional approach, which can be observed through histogram and variogram validations. Unlike conventional cosimulation, the integration of inverse transform sampling helps to reproduce an inequality constraint in the bivariate relationship and increases the correlation coefficient.
Next, a two-stage stochastic production scheduling approach is used as a replacement for deterministic mine planning. In the first stage, extraction periods are obtained using an e-type model and remain fixed for all realizations. The second stage decisions re-evaluate block destinations based on a set of realization. This approach minimizes the risk of sending an extracted material to the wrong destinations while producing optimal production schedules. Its NPV is only 17% lower than the best possible NPV for each scenario and 59% higher than the deterministic plan. Moreover, the proposed cosimulation algorithm increases the NPV by 0.76% due to the reproduction of an inequality constraint.

This study aimed to evaluate the temporal change and accuracy of interpolation techniques used for spatial zonation of two groundwater quantity parameters including water table and depth to water table over 11 years. The study was conducted based on the data collected from piezometric wells of Sari-Neka Plain in Mazandaran Province, Iran. The investigated methods included a set of geostatistical approaches involving simple Kriging, ordinary Kriging, Radial Basis Function (RBF), and a deterministic interpolation method called Inverse Distance Weighting (IDW) with powers of 1 and 5. Subsequent to quality control and data normalization, the most appropriate variogram was chosen based on low RSS and high r 2 while the most suitable interpolation technique was determined regarding the cross validation, Mean Absolute Error (MAE), and Mean Bias Error (MBE). The results demonstrated that Simple Kriging was the most suitable method for zoning the depth to groundwater over the years 2001, 2006, and 2012. Meanwhile, the most suitable methods for zoning the water table included IDW with a power of 1for the year 2001, RBF for the year 2006, and IDW with a power of 5 for the year 2012. The important finding was that the interpolation methods showed a lower error for estimating water table than estimating depth to groundwater. This study also revealed a drop in water table in the study area over the 11 years' period. Meanwhile, new water table classes have been added and extended between the years 2006 and 2012 that had not existed five years earlier. The highest water table losses were observed in three points at 13m depth to water table in the middle and northern parts of the study area.

Detailed information on soil to manage polluted or agricultural sites is often prohibitively expensive to obtain. To sample adequately, the approximate scale of spatial variation needs to be known. If soil data are available, variograms can be computed and used to determine the kriging errors for several grid intervals and an interval selected to meet a specific tolerable error. In the absence of prior knowledge, if soil properties appear related to ancillary data such as aerial photographs, elevation or apparent electrical conductivity (ECa), individual or multivariate variograms of such data may indicate the scale of variation in the soil. If the scale of variation indicates too few data to compute a reliable variogram conventionally, it can be stimated by residual maximum likelihood or a standardized variogram from ancillary data can be used.

The fractal dimension (D HB) is an interesting metrics because it is supposed to quantify by a single value, scale independence and roughness of ecological objects. However, we show here that those two properties may be quantified by a single dimension only in some specific cases. In general, a non-integer D HB quantifies only the roughness, and self-similarity needs to be evidenced or postulated by other means. Second, we revisit some aspects of the practical estimation of D HB. We recommend the use of madogram instead of variogram for estimations based on geostatistics. We propose a simplification of its estimation for 2D fields and discuss its possible relationship with self-similarity. We finally underline the problem of scale and resolution. Field data recorded during a scientific acoustic survey on the North Sea herring are used for illustrations. The paper concludes on a synthesis of practical recommendations to ecologists when using fractal dimension.

Analysis of the spatial pattern is an important element for the optimum management and having knowledge about ecosystem function. This study was conducted to analyze the spatial pattern of trees and shrub in the riparian forest landscape of Karkhe, also for assessment of their spatial variability regarding the distance from the river. The number of trees and shrubs were sampled using 117 plot (20m × 20 m) along parallel transects (perpendicular to the river). The distance between transects was 500 m. The variograms of Populus euphratica and Lycium shawii were spherical. Tamarix sp. and total of them were exponential. All revealed the presence of spatial autocorrelation. The range of influence was 186 m for Tamarix sp., 362 m for Populus euphratica, 749 m for Lycium shawii and 208 for total. The kriging maps showed spatial variability of them. The spatial pattern of species total and Tamarix sp. are similar that was shown in correlation. So Tamarix sp. is a major part of this forest. Populus euphratica is replacing with Lycium shawii. Also the higher distribution is nearby river for all trees and shrub.
Extended Abstract
1-Introduction
In the last 15-20 years, riparian forests have become recognized as important components of landscape and serve as a vital link between the aquatic environment and upland ecosystems. Riparian ecosystems are aquatic-terrestrial ecotones with unique biotic, biophysical and landscape characteristics. Accordingly, sustainability and maintenance of riparian vegetation or restoring of degraded sites is critical to sustain inherent ecosystem function and values. Many scientists believe promoting sustainability is the overarching goal of landscape (and regional) planning. A current challenge in ecology is underestimating how patterns and processes vary with scale and searching for general principles (assembly rules) which determine the species composition of communities. Also a fundamental and unanswered research question in landscape ecology centers on how the spatial arrangement of the ecosystems influences the distribution and abundance of species. Description of patterns in species assemblages and diversity is an essential step before generating hypotheses in functional ecology. Information about the spatial pattern of tree species in riparian forest ecosystems is limited though required, e.g. for understanding spatial distribution effects of trees on ecosystem processes. The practical consequences of these findings are useful for sustainable management of forest ecosystems and in monitoring of forest evolution. One main reason responsible for the absence of information about spatial pattern researche is the lack of adequate methods for analyzing data. Geostatistics provide descriptive tools such as variogram to characterize the spatial pattern of continuous and categorical forest attributes. This method allows assessment of consistency of spatial patterns as well as the scale at which they are expressed. This study was conducted to analyze spatial patterns of tree species in the riparian forest landscape of Karkhe.
2- Materials and Methods
The study was carried out in Wildlife Refugee of Karkhe in the riparian forest of the southwestern Iran (31o  57/- 32 o 05/ N and 48 o 13/- 48 o 16/ E). The climate of the study area is semi-arid. Average yearly rainfall is about 325.8 mm with a mean temperature of 24oc. Plant cover, mainly comprises Populus euphratica and Tamarix sp. The both sides of river are similar, so we sampled on one of the two sides. The number of trees and shrubs were sampled using 117 plot (20m × 20 m) along parallel transects (perpendicular to the river). The maximum distance between plots was 0.5 km. The sampling procedure was hierarchically, we considered maximum distance between samples as 0.5 km, but the samples was taken at 250m, 100m, 50m at different location of sampling.We began with an exploratory analysis of patterns in the data. So, classical statistical parameters, i.e. mean, standard deviation, coefficient of variation, minimum and maximum, were calculated using SPSS17 software. Next, we applied geostatistics methods to examine spatial structure in tree distribution.To determining the spatial structure, we calculated the semivariances. Semivariance quantifies the spatial dependence of spatially ordered variable values. The semivariance essentially expresses the average variance of pairs of points at a given distance. Empirical variograms are plots of the semivariances, averaged over distance classes (called lags), against the lag distance. To describe spatial autocorrelation of a variable quantitatively, a (theoretical) variogram model can be fitted to the empirical variogram in order to obtain the model parameters nugget, sill and range. 
3- Results and Discussion
The variograms of Populus euphratica and Lycium shawii were spherical. Tamarix sp. and total of them were exponential. All revealed the presence of spatial autocorrelation. The range of influence was 186 m for Tamarix sp., 362 m for Populus euphratica, 749 m for Lycium shawii and 208 for total. The kriging maps showed spatial variability of them. Typically these structures constitute one source of the nugget variance of the variograms (Rossi, 2003). However, the variograms reported here featured a somewhat high ratio of nugget variance to sill. This result showed that there was the small-scale variability and important proportion of unexplained variance. Spatial distribution of tree speciec may be influenced by factors like gradients in soil organic matter (quantity and or quality) and texture. These factors together with intrinsic population processes constitute proximate controlling factors of population structure.
4- Conclusion
The spatial pattern of total species and Tamarix sp. are similar that was shown in correlation. So Tamarix sp. is a major part of this forest. Populus euphratica is replacing with Lycium shawii. Also the higher distribution is nearby river for all trees and shrub.